We next consider ultracold spin-exchange collisions of \(^{87}\) Rb atoms, which have been the subject of much experimental study (for a review, see [1]}).
The two-atom threshold states \(|j_1m_{1}\rangle |j_2 m_{2}\rangle = |F_1m_{F_1}\rangle |F_2 m_{F_2}\rangle \) , where \(|j_i m_{F_i}\rangle = F_i m_{F_i}\rangle \)  are the atomic hyperfine states and \(\mathbf {j}_i=\mathbf {S}_i+\mathbf {I}_i\)  are the total atomic angular momenta, which are vector sums of the electron and nuclear spins of the \(i\) -th atom (\(i=1,2\) ). The quantum scattering problem for \(^{87}\) Rb + \(^{87}\) Rb is solved using the standard CC approach as described in, e.g., Ref. [2]} by integrating the CC equations on a grid of \(R\)  values from 2 to 400\(a_0\)  with a step size of \(5 \times 10^{-3} a_0\) .
